The Enumerative Geometry of K3 Surfaces and Modular Forms
نویسندگان
چکیده
Let X be a K3 surface and C be a holomorphic curve in X representing a primitive homology class. We count the number of curves of geometric genus g with n nodes passing through g generic points in X in the linear system |C| for any g and n satisfying C · C = 2g + 2n − 2. When g = 0, this coincides with the enumerative problem studied by Yau and Zaslow who obtained a conjectural generating function for the numbers. Recently, Göttsche has generalized their conjecture to arbitrary g in terms of quasi-modular forms. We prove these formulas using Gromov-Witten invariants for families, a degeneration argument, and an obstruction bundle computation. Our methods also apply to P blown up at 9 points where we show that the ordinary Gromov-Witten invariants of genus g constrained to g points are also given in terms of quasi-modular forms.
منابع مشابه
The Enumerative Geometry of K3 Surfaces and Modular Forms Jim Bryan and Naichung Conan Leung
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